Absolute cross sections and rate coefficients were measured for the dissociative recombination of CN+(X (1)Sigma(+) and a (3)Pi, nu=0) with electrons using the heavy ion storage-ring CRYRING ht Stockholm University. The a (3)Pi State that lies only 0.08 eV above the X (1)Sigma(+) ion groundstate is most certainly populated in the ion beam injected into CRYRING, as this is a long-lived excited state. Over the 1 meV-3 eV energy range that was investigated, two energy dependencies were found for the cross sections, namely E-105 below 0.1 eV and E-1.41 above 0.4 eV. The energy domain 0.1-0.3 eV exhibits a resonant structure. Branching fractions over the final atomic product states were determined using a position sensitive imaging detector. Thermal rate coefficients are deduced from the measurements and the efficiency displayed by the dissociative recombination mechanism is typical of that of a diatomic ion (3.4 x 10(-7) cm(3) s(-1) at room temperature). Cross sections for the dissociative excitation process are also reported and their magnitude at the plateau are unusually large. This may be explained by the number of potential curves which can be populated at moderate collision energy (<10 eV) for the CN+ ion.